Classification of compact ancient solutions to the curve shortening flow

By P. Daskalopoulos, R. Hamilton, N. Šešum. 2008

121 citations recorded by the source corpus.

We consider an embedded convex ancient solution $\Gamma_t$ to the curve shortening flow in $\mathbb{R}^2$. We prove that there are only two possibilities: the family $\Gamma_t$ is either the family of contracting circles, which is a type I ancient solution, or the family of evolving Angenent ovals, which correspond to a type II ancient solution to the curve shortening flow. We also give a necessary and sufficient curvature condition for an embedded, closed ancient solution to the curve shortening flow to be convex.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f36ab7d23049703732ead80315c524ce24ead21c

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Convex ancient solutions of the mean curvature flow (2014, background): “Compact convex ancient solutions in the plane have been completely classified by DaskalopoulosHamilton-Sesum [7] to be either shrinking round circles or an Angenent oval.”
Ancient asymptotically cylindrical flows and applications (2019, background): “In [DHS10], Daskalopoulos, Hamilton and Sesum showed that closed embedded ancient evolutions of curves are either a family of round shrinking circles or Angenent’s paper-clip solution.”
Unique asymptotics of ancient convex mean curvature flow solutions (2015, background): “N. Sesum thanks the NSF for support in DMS-1056387.”
Ancient solutions of the mean curvature flow (2013, background): “Daskalopoulos, Hamilton and Sesum have obtained a complete classification of ancient solutions in the case of closed embedded curves [6].”
Mean Curvature Flow of Mean Convex Hypersurfaces (2013, result): “Then, by results of Huisken and Sinestrari [26] and Daskalopoulos, Hamilton, and Sesum [10], it must be a round sphere.”
Uniqueness of compact ancient solutions to three-dimensional Ricci flow (2020, background): “In [12], the authors classified compact, convex ancient solutions to the curve shortening flow.”
Uniqueness of two-convex closed ancient solutions to the mean curvature flow (2018, background): “In previous classifications of ancient solutions to mean curvature flow and Ricci flow, [8], [9], [6, 7], an essential role in the proofs was played by the fact that all such solutions were given in closed form or they were solitons.”